LMFDB - Embedded newform 2304.4.c.a.2303.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,4,Mod(2303,2304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2304.2303"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-36,0,0,0,0,0,0,0,0,0,0,0,-426] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$135.940400653$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2303.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	2304.2303
Dual form			2304.4.c.a.2303.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	18.3848i	1.64438i	0.569210	+	0.822192i	$0.307249\pi$
$5$	18.3848i	1.64438i	−0.569210	+	0.822192i	$0.692751\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−18.0000	−0.384023	−0.192012	−	0.981393i	$-0.561501\pi$
$13$	−18.0000	−0.384023	−0.192012	+	0.981393i	$0.561501\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	140.007i	1.99745i	0.0504408	+	0.998727i	$0.483937\pi$
$17$	140.007i	1.99745i	−0.0504408	+	0.998727i	$0.516063\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−213.000	−1.70400
$26$	0	0
$27$	0	0
$28$	0	0
$29$	108.894i	0.697282i	0.937256	+	0.348641i	$0.113357\pi$
$29$	108.894i	0.697282i	−0.937256	+	0.348641i	$0.886643\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−396.000	−1.75951	−0.879757	−	0.475424i	$-0.842295\pi$
$37$	−396.000	−1.75951	−0.879757	+	0.475424i	$0.842295\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 496.389i	− 1.89080i	−0.325908	−	0.945402i	$-0.605670\pi$
$41$	− 496.389i	− 1.89080i	0.325908	−	0.945402i	$-0.394330\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	770.746i	1.99755i	0.0494806	+	0.998775i	$0.484243\pi$
$53$	770.746i	1.99755i	−0.0494806	+	0.998775i	$0.515757\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−468.000	−0.982316	−0.491158	−	0.871071i	$-0.663426\pi$
$61$	−468.000	−0.982316	−0.491158	+	0.871071i	$0.663426\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 330.926i	− 0.631482i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	592.000	0.949156	0.474578	−	0.880214i	$-0.342601\pi$
$73$	592.000	0.949156	0.474578	+	0.880214i	$0.342601\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−2574.00	−3.28458
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1056.42i	1.25820i	0.777323	+	0.629101i	$0.216577\pi$
$89$	1056.42i	1.25820i	−0.777323	+	0.629101i	$0.783423\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1816.00	−1.90090	−0.950448	−	0.310884i	$-0.899375\pi$
$97$	−1816.00	−1.90090	−0.950448	+	0.310884i	$0.899375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	948.937i	0.934879i	0.884025	+	0.467440i	$0.154823\pi$
$101$	948.937i	0.934879i	−0.884025	+	0.467440i	$0.845177\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	1746.00	1.53428	0.767140	−	0.641480i	$-0.221679\pi$
$109$	1746.00	1.53428	0.767140	+	0.641480i	$0.221679\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2354.67i	− 1.96025i	−0.198380	−	0.980125i	$-0.563568\pi$
$113$	− 2354.67i	− 1.96025i	0.198380	−	0.980125i	$-0.436432\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1617.86i	− 1.15765i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 827.315i	− 0.515929i	−0.966154	−	0.257965i	$-0.916948\pi$
$137$	− 827.315i	− 0.515929i	0.966154	−	0.257965i	$-0.0830518\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2002.00	−1.14660
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3149.45i	1.73163i	0.500362	+	0.865816i	$0.333200\pi$
$149$	3149.45i	1.73163i	−0.500362	+	0.865816i	$0.666800\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3924.00	1.99471	0.997354	−	0.0726920i	$-0.0231590\pi$
$157$	3924.00	1.99471	0.997354	+	0.0726920i	$0.0231590\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−1873.00	−0.852526
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4309.11i	− 1.89373i	−0.321630	−	0.946866i	$-0.604231\pi$
$173$	− 4309.11i	− 1.89373i	0.321630	−	0.946866i	$-0.395769\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−3942.00	−1.61882	−0.809410	−	0.587243i	$-0.800213\pi$
$181$	−3942.00	−1.61882	−0.809410	+	0.587243i	$0.800213\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 7280.37i	− 2.89332i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	5362.00	1.99982	0.999910	−	0.0134266i	$-0.00427395\pi$
$193$	5362.00	1.99982	0.999910	+	0.0134266i	$0.00427395\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4651.35i	− 1.68221i	−0.540874	−	0.841104i	$-0.681906\pi$
$197$	− 4651.35i	− 1.68221i	0.540874	−	0.841104i	$-0.318094\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9126.00	3.10921
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2520.13i	− 0.767069i
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	6390.00	1.84394	0.921972	−	0.387257i	$-0.126577\pi$
$229$	6390.00	1.84394	0.921972	+	0.387257i	$0.126577\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5434.82i	− 1.52810i	−0.645158	−	0.764050i	$-0.723208\pi$
$233$	− 5434.82i	− 1.52810i	0.645158	−	0.764050i	$-0.276792\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−5272.00	−1.40913	−0.704563	−	0.709641i	$-0.748857\pi$
$241$	−5272.00	−1.40913	−0.704563	+	0.709641i	$0.748857\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6305.98i	1.64438i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6809.44i	− 1.65277i	−0.563108	−	0.826383i	$-0.690394\pi$
$257$	− 6809.44i	− 1.65277i	0.563108	−	0.826383i	$-0.309606\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−14170.0	−3.28474
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8164.25i	− 1.85050i	−0.379363	−	0.925248i	$-0.623857\pi$
$269$	− 8164.25i	− 1.85050i	0.379363	−	0.925248i	$-0.376143\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9126.00	−1.97952	−0.989762	−	0.142727i	$-0.954413\pi$
$277$	−9126.00	−1.97952	−0.989762	+	0.142727i	$0.954413\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1158.24i	0.245889i	0.992414	+	0.122945i	$0.0392337\pi$
$281$	1158.24i	0.245889i	−0.992414	+	0.122945i	$0.960766\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14689.0	−2.98982
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 4218.60i	− 0.841137i	−0.907261	−	0.420569i	$-0.861831\pi$
$293$	− 4218.60i	− 0.841137i	0.907261	−	0.420569i	$-0.138169\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8604.08i	− 1.61530i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6838.00	1.23485	0.617423	−	0.786632i	$-0.288177\pi$
$313$	6838.00	1.23485	0.617423	+	0.786632i	$0.288177\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10571.2i	1.87300i	0.350670	+	0.936499i	$0.385954\pi$
$317$	10571.2i	1.87300i	−0.350670	+	0.936499i	$0.614046\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3834.00	0.654376
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−416.000	−0.0672432	−0.0336216	−	0.999435i	$-0.510704\pi$
$337$	−416.000	−0.0672432	−0.0336216	+	0.999435i	$0.510704\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−8964.00	−1.37488	−0.687438	−	0.726243i	$-0.741265\pi$
$349$	−8964.00	−1.37488	−0.687438	+	0.726243i	$0.741265\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 11416.9i	− 1.72142i	−0.509092	−	0.860712i	$-0.670019\pi$
$353$	− 11416.9i	− 1.72142i	0.509092	−	0.860712i	$-0.329981\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6859.00	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10883.8i	1.56078i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6372.00	−0.884530	−0.442265	−	0.896884i	$-0.645825\pi$
$373$	−6372.00	−0.884530	−0.442265	+	0.896884i	$0.645825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1960.10i	− 0.267773i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10582.6i	1.37932i	0.724131	+	0.689662i	$0.242241\pi$
$389$	10582.6i	1.37932i	−0.724131	+	0.689662i	$0.757759\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12564.0	−1.58834	−0.794168	−	0.607699i	$-0.792093\pi$
$397$	−12564.0	−1.58834	−0.794168	+	0.607699i	$0.792093\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 9533.21i	− 1.18720i	−0.804761	−	0.593598i	$-0.797707\pi$
$401$	− 9533.21i	− 1.18720i	0.804761	−	0.593598i	$-0.202293\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14920.0	−1.80378	−0.901891	−	0.431964i	$-0.857821\pi$
$409$	−14920.0	−1.80378	−0.901891	+	0.431964i	$0.857821\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	10890.0	1.26068	0.630340	−	0.776319i	$-0.282916\pi$
$421$	10890.0	1.26068	0.630340	+	0.776319i	$0.282916\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 29821.5i	− 3.40366i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−4862.00	−0.539614	−0.269807	−	0.962914i	$-0.586960\pi$
$433$	−4862.00	−0.539614	−0.269807	+	0.962914i	$0.586960\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−19422.0	−2.06897
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4238.40i	0.445484i	0.974877	+	0.222742i	$0.0715008\pi$
$449$	4238.40i	0.445484i	−0.974877	+	0.222742i	$0.928499\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10456.0	−1.07026	−0.535132	−	0.844768i	$-0.679738\pi$
$457$	−10456.0	−1.07026	−0.535132	+	0.844768i	$0.679738\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12262.6i	1.23889i	0.785040	+	0.619445i	$0.212642\pi$
$461$	12262.6i	1.23889i	−0.785040	+	0.619445i	$0.787358\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	7128.00	0.675694
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 33386.8i	− 3.12580i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−15246.0	−1.39279
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−17446.0	−1.53730
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22815.5i	1.98680i	0.114715	+	0.993398i	$0.463404\pi$
$509$	22815.5i	1.98680i	−0.114715	+	0.993398i	$0.536596\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 17831.8i	− 1.49947i	−0.661736	−	0.749737i	$-0.730180\pi$
$521$	− 17831.8i	− 1.49947i	0.661736	−	0.749737i	$-0.269820\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8935.00i	0.726112i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5922.00	0.470622	0.235311	−	0.971920i	$-0.424389\pi$
$541$	5922.00	0.470622	0.235311	+	0.971920i	$0.424389\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	32099.8i	2.52295i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11462.2i	− 0.871937i	−0.899962	−	0.435969i	$-0.856406\pi$
$557$	− 11462.2i	− 0.871937i	0.899962	−	0.435969i	$-0.143594\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	43290.0	3.22341
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21981.1i	1.61950i	0.586774	+	0.809751i	$0.300398\pi$
$569$	21981.1i	1.61950i	−0.586774	+	0.809751i	$0.699602\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3454.00	−0.249206	−0.124603	−	0.992207i	$-0.539766\pi$
$577$	−3454.00	−0.249206	−0.124603	+	0.992207i	$0.539766\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 28192.3i	− 1.95231i	−0.217070	−	0.976156i	$-0.569650\pi$
$593$	− 28192.3i	− 1.95231i	0.217070	−	0.976156i	$-0.430350\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	17030.0	1.15585	0.577927	−	0.816089i	$-0.303862\pi$
$601$	17030.0	1.15585	0.577927	+	0.816089i	$0.303862\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 24470.1i	− 1.64438i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	19548.0	1.28799	0.643994	−	0.765031i	$-0.277276\pi$
$613$	19548.0	1.28799	0.643994	+	0.765031i	$0.277276\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7776.76i	0.507424i	0.967280	+	0.253712i	$0.0816515\pi$
$617$	7776.76i	0.507424i	−0.967280	+	0.253712i	$0.918348\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3119.00	0.199616
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 55442.8i	− 3.51455i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6174.00	−0.384023
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 30916.1i	− 1.90501i	−0.304518	−	0.952507i	$-0.598495\pi$
$641$	− 30916.1i	− 1.90501i	0.304518	−	0.952507i	$-0.401505\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24303.3i	1.45645i	0.685340	+	0.728224i	$0.259654\pi$
$653$	24303.3i	1.45645i	−0.685340	+	0.728224i	$0.740346\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	22068.0	1.29856	0.649278	−	0.760551i	$-0.275071\pi$
$661$	22068.0	1.29856	0.649278	+	0.760551i	$0.275071\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	4462.00	0.255568	0.127784	−	0.991802i	$-0.459214\pi$
$673$	4462.00	0.255568	0.127784	+	0.991802i	$0.459214\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27612.5i	1.56756i	0.621041	+	0.783778i	$0.286710\pi$
$677$	27612.5i	1.56756i	−0.621041	+	0.783778i	$0.713290\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	15210.0	0.848386
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 13873.4i	− 0.767106i
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	69498.0	3.77679
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 7935.15i	− 0.427541i	−0.976884	−	0.213771i	$-0.931425\pi$
$701$	− 7935.15i	− 0.427541i	0.976884	−	0.213771i	$-0.0685745\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36810.0	−1.94983	−0.974914	−	0.222580i	$-0.928552\pi$
$709$	−36810.0	−1.94983	−0.974914	+	0.222580i	$0.928552\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 23194.5i	− 1.18817i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	38718.0	1.95100	0.975499	−	0.220003i	$-0.0706066\pi$
$733$	38718.0	1.95100	0.975499	+	0.220003i	$0.0706066\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−57902.0	−2.84747
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35046.0	1.68265	0.841327	−	0.540527i	$-0.181775\pi$
$757$	35046.0	1.68265	0.841327	+	0.540527i	$0.181775\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41862.1i	1.99409i	0.0768304	+	0.997044i	$0.475520\pi$
$761$	41862.1i	1.99409i	−0.0768304	+	0.997044i	$0.524480\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−9650.00	−0.452520	−0.226260	−	0.974067i	$-0.572650\pi$
$769$	−9650.00	−0.452520	−0.226260	+	0.974067i	$0.572650\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16044.3i	0.746535i	0.927724	+	0.373268i	$0.121763\pi$
$773$	16044.3i	0.746535i	−0.927724	+	0.373268i	$0.878237\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	72141.9i	3.28007i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8424.00	0.377232
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18157.1i	− 0.806972i	−0.914986	−	0.403486i	$-0.867798\pi$
$797$	− 18157.1i	− 0.806972i	0.914986	−	0.403486i	$-0.132202\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11620.6i	0.505016i	0.967595	+	0.252508i	$0.0812555\pi$
$809$	11620.6i	0.505016i	−0.967595	+	0.252508i	$0.918745\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34550.7i	1.46873i	0.678756	+	0.734364i	$0.262520\pi$
$821$	34550.7i	1.46873i	−0.678756	+	0.734364i	$0.737480\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−23166.0	−0.970553	−0.485276	−	0.874361i	$-0.661281\pi$
$829$	−23166.0	−0.970553	−0.485276	+	0.874361i	$0.661281\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	48022.4i	1.99745i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	12531.0	0.513797
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 34434.7i	− 1.40188i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−45468.0	−1.82508	−0.912541	−	0.408986i	$-0.865883\pi$
$853$	−45468.0	−1.82508	−0.912541	+	0.408986i	$0.865883\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18340.9i	0.731055i	0.930800	+	0.365528i	$0.119111\pi$
$857$	18340.9i	0.731055i	−0.930800	+	0.365528i	$0.880889\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	79222.0	3.11402
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29844.0	−1.14910	−0.574550	−	0.818470i	$-0.694823\pi$
$877$	−29844.0	−1.14910	−0.574550	+	0.818470i	$0.694823\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31578.0i	1.20759i	0.797139	+	0.603796i	$0.206346\pi$
$881$	31578.0i	1.20759i	−0.797139	+	0.603796i	$0.793654\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−107910.	−3.99001
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 72472.8i	− 2.66196i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	84348.0	2.99821
$926$	0	0
$927$	0	0
$928$	0	0
$929$	11747.9i	0.414893i	0.978246	+	0.207446i	$0.0665152\pi$
$929$	11747.9i	0.414893i	−0.978246	+	0.207446i	$0.933485\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51946.0	−1.81110	−0.905551	−	0.424238i	$-0.860542\pi$
$937$	−51946.0	−1.81110	−0.905551	+	0.424238i	$0.860542\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	56454.0i	1.95574i	0.209223	+	0.977868i	$0.432907\pi$
$941$	56454.0i	1.95574i	−0.209223	+	0.977868i	$0.567093\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−10656.0	−0.364498
$950$	0	0
$951$	0	0
$952$	0	0
$953$	51102.6i	1.73702i	0.495676	+	0.868508i	$0.334921\pi$
$953$	51102.6i	1.73702i	−0.495676	+	0.868508i	$0.665079\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29791.0	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	98579.2i	3.28847i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 56244.7i	− 1.84179i	−0.389813	−	0.920894i	$-0.627460\pi$
$977$	− 56244.7i	− 1.84179i	0.389813	−	0.920894i	$-0.372540\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	85514.0	2.76620
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−34164.0	−1.08524	−0.542620	−	0.839978i	$-0.682568\pi$
$997$	−34164.0	−1.08524	−0.542620	+	0.839978i	$0.682568\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.c.a.2303.2		2
3.2	odd	2	inner	2304.4.c.a.2303.1		2
4.3	odd	2	CM	2304.4.c.a.2303.2		2
8.3	odd	2		2304.4.c.c.2303.1		2
8.5	even	2		2304.4.c.c.2303.1		2
12.11	even	2	inner	2304.4.c.a.2303.1		2
16.3	odd	4		1152.4.f.c.575.3	yes	4
16.5	even	4		1152.4.f.c.575.2	yes	4
16.11	odd	4		1152.4.f.c.575.2	yes	4
16.13	even	4		1152.4.f.c.575.3	yes	4
24.5	odd	2		2304.4.c.c.2303.2		2
24.11	even	2		2304.4.c.c.2303.2		2
48.5	odd	4		1152.4.f.c.575.4	yes	4
48.11	even	4		1152.4.f.c.575.4	yes	4
48.29	odd	4		1152.4.f.c.575.1	✓	4
48.35	even	4		1152.4.f.c.575.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.f.c.575.1	✓	4	48.29	odd	4
1152.4.f.c.575.1	✓	4	48.35	even	4
1152.4.f.c.575.2	yes	4	16.5	even	4
1152.4.f.c.575.2	yes	4	16.11	odd	4
1152.4.f.c.575.3	yes	4	16.3	odd	4
1152.4.f.c.575.3	yes	4	16.13	even	4
1152.4.f.c.575.4	yes	4	48.5	odd	4
1152.4.f.c.575.4	yes	4	48.11	even	4
2304.4.c.a.2303.1		2	3.2	odd	2	inner
2304.4.c.a.2303.1		2	12.11	even	2	inner
2304.4.c.a.2303.2		2	1.1	even	1	trivial
2304.4.c.a.2303.2		2	4.3	odd	2	CM
2304.4.c.c.2303.1		2	8.3	odd	2
2304.4.c.c.2303.1		2	8.5	even	2
2304.4.c.c.2303.2		2	24.5	odd	2
2304.4.c.c.2303.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+18.3848i q^{5} -18.0000 q^{13} +140.007i q^{17} -213.000 q^{25} +108.894i q^{29} -396.000 q^{37} -496.389i q^{41} +343.000 q^{49} +770.746i q^{53} -468.000 q^{61} -330.926i q^{65} +592.000 q^{73} -2574.00 q^{85} +1056.42i q^{89} -1816.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 36 q^{13} - 426 q^{25} - 792 q^{37} + 686 q^{49} - 936 q^{61} + 1184 q^{73} - 5148 q^{85} - 3632 q^{97}+O(q^{100}) \) 2 * q - 36 * q^13 - 426 * q^25 - 792 * q^37 + 686 * q^49 - 936 * q^61 + 1184 * q^73 - 5148 * q^85 - 3632 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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